Average Error: 23.8 → 6.0
Time: 3.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{\sqrt{\beta + \alpha}}{\sqrt{\sqrt[3]{\left(\left(2.0 + \beta\right) + i \cdot 2\right) + \alpha}}} \cdot \frac{\sqrt{\beta + \alpha}}{\left|\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\right|}\right) \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}} + 1.0}{2.0} \le 2.8305731925196304 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha}}{\alpha} - \left(\frac{4.0}{\alpha} - 2.0\right)}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{\beta - \alpha}}}{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (* (/ (sqrt (+ alpha beta)) (fabs (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (/ (sqrt (+ beta alpha)) (sqrt (cbrt (+ alpha (+ (* i 2) (+ beta 2.0))))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0) < 2.8305731925196304e-16

    1. Initial program 62.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 28.5

      \[\leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified28.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{8.0}{\alpha}}{\alpha} - \left(\frac{4.0}{\alpha} - 2.0\right)}{\alpha}}}{2.0}\]

    if 2.8305731925196304e-16 < (/ (+ (* (* (/ (sqrt (+ alpha beta)) (fabs (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (/ (sqrt (+ beta alpha)) (sqrt (cbrt (+ alpha (+ (* i 2) (+ beta 2.0))))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0)

    1. Initial program 14.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied associate-/l*0.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{\sqrt{\beta + \alpha}}{\sqrt{\sqrt[3]{\left(\left(2.0 + \beta\right) + i \cdot 2\right) + \alpha}}} \cdot \frac{\sqrt{\beta + \alpha}}{\left|\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\right|}\right) \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}} + 1.0}{2.0} \le 2.8305731925196304 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha}}{\alpha} - \left(\frac{4.0}{\alpha} - 2.0\right)}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{\beta - \alpha}}}{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018215 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))